44 XII TOPOLOGY AND TOPOLOGICAL ALGEBRA (d) Show that every closed subgroup H of Tp , other than Tp and {0}, is totally dis- connected and isomorphic to a group of the form Z/nZ or (Z/nZ) x Zp, where n is an integer prime to p. (Consider the groups /n(H).) (e) Show that Tp is an indecomposable compact connected space, i.e., that it cannot be covered by two compact connected sets P, Q, each distinct from Tp . (Observe that there exists an integer n such that /n(P) ^ G« and /n(Q) =Ł Gn , and examine the sets /n+i(P) and/w+1(Q) to obtain a contradiction.) 6. A topological group G is said to have no small subgroups if there exists a neighborhood V of e such that {e} is the only subgroup of G contained in V. Let G be a locally compact metrizable group* with no small subgroups. (a) Show that there exists a compact neighborhood V of e such that, for all x, y in V, the relation x2 y2 implies x = y. (We may assume that G is not commutative. If the result were false, there would exist two sequences (*), (yn) of points of G, both tending to et such that x2 = y2 and xny* 1 = an ^ e. Let U be a compact symmetric neighborhood of e not containing any subgroup of G other than {e}, and let pn be the smallest integer p>Q such that a%+1 <ŁU. Show, by passing to a subsequence, that we may assume that a = lim a*" exists, is „=e and belongs to U. Show that /I-*- op a"1 = a and so obtain a contradiction.) (b) Let U be a compact symmetric neighborhood of e, containing no subgroup other than {e}, and let V be a neighborhood of e. Show that there exists a number c(V) > 0 such that, whenever p and q are strictly positive integers such that p :g c(V)q and x e G is such that x, x2, . . . , x* are in U, then xp e V. (Argue by contradiction: suppose that there exist two sequences of integers (jpn), (<?) such that lim pn/qn 0, and for H-K» each n an element an e G such thataj e U for 1 <J h <* qnt but apn" $ V. We may also assume that the sequence (apnn) has a limit a ^ e such that a e U. Show that am e U for all m > 0, and hence get a contradiction.) 7. Let G be a locally compact metrizable group with no small subgroups, and let V be a compact symmetric neighborhood of e containing no subgroup of G other than {e}, and such that for all x, y e V the relation x2 = y2 implies x = y (Problem 6(a)). (a) Show that if (an) is any sequence of points of V with e as limit, there exists a subsequence (bn) of (an) and a sequence (kn) of integers >0 such that the sequence (6jn) converges to a point ^e. (Consider the smallest of the integers k such that (b) Show that if r, s are two real numbers such that the sequences (6^r*n3), (b^) converge to x, y, respectively, in G, then the sequence (b^r+5)kn^) converges to xy. (Here [t] denotes the integral part of the real number t.) (c) Using (a), (b), and the uniqueness of the square root, show that for every dyadic number re [0, 1] the sequence (bljkn*) converges in G. (d) Let W ^ V be a neighborhood of e in G. Show that, for every real number r e [0, 1], there exists a dyadic number s such that b^r+s)kn'J e W for all n (use Problem 6(b)). Deduce that, for each r e [0, 1], the sequence (b^) has a limit X(r). (e) If -1 <; r <; 0, put X(r) = (X(-r))-1- Show that, if r, s and r + s are all in the interval [ 1, 1], we have X(r)X(s) = X(r + s) and that the mapping r h->X(r) of [1, 1] into G is continuous. Deduce that this mapping can be extended to a non- constant continuous homomorphism of R into G. 8. Let I be the compact interval [0, 1 ] in R, and let G be the group of all homeomorphisms of I onto I, which is contained in the Banach space 3fR(I) (7.2.1). Show that thevery set consisting of a single point is closed,